2. Passage of Radiation
Through Matter
Passage of Radiation Through
Matter: Contents
• Energy Loss of Heavy Charged Particles by Atomic
Collision (addendum)
• Cherenkov Radiation
• Energy loss of Electrons and Positrons
• Multiple Coulomb Scattering
• The Interaction of Photons:
•
•
•
•
Photo electric effect
Compton scattering
Pair production
Electron-photon shower etc.
• The Interaction of Neutrons
Relativistic Variables
v
β=
c
E
γ =
2 =
Mc
p
=
Mc
Velocity in units
of speed of light
1
1− β
γ −1
2
2
Lorentz factor
Addendum: 2.1
Energy Loss of
Heavy Charged
Particles
MIP @
At around
β≈0.96
Most relativistic
particles
considered MIPs
Addendum: 2.1 Energy Loss of
Heavy Charged Particles (cont.):
Chemical Compounds and
Mixtures
If accurate values are desired: direct measurements,
but good approximation is given by the
Bragg-Kleeman Rule:
1 dE
=
ρ dx
∑
wi: weight of element i or atomic fraction
ρ: atomic density
Mass stopping power
i
wi  dE 


ρi  dx  i
More explicitly: wi=aiAi/Am,
Ai: atomic weight, Am=ΣaiAi
ai: Number of atoms of the ith
element in the molecule M
Addendum: 2.1 Energy Loss of
Heavy Charged Particles (cont.):
Channeling
• Important exception to Bethe-Bloch formula
•
•
•
•
•
•
Occurs in materials having a spatially symmetric atomic structure,
i.e. crystals
Only occurs at an incident angle θ < certain critical angle w.r.t. a
symmetry axes of the crystal θc
Particle suffers a series of correlated small-angle scatterings:
Slowly oscillating trajectory, wavelength of the trajectory is
generally many lattice lengths long
Channeling greatly reduces energy loss (particle encounters less
electrons than it normally would)
In general: θc ≈ 1o for β≈0.1
•
Addendum: 2.1 Energy Loss of
Heavy Charged Particles (cont.):
Range
Range: penetration depth/distance of a particle in a material before it
loses all its energy
• well defined number
• Same for all identical particles with the same initial energy in the same
type of material: R( Material, Particle type, E)
• Experimental determination: passing a beam of monoenergetic particles
through different thicknesses of material
→ measure ratio of transmitted to incident particles → range number
distance curve
•
Spread of ranges because energy loss is not continuous but statistical in
nature → range straggling
Addendum: 2.1 Energy
Loss of Heavy Charged
Particles (cont.): Range
Tmin:
minimum energy at which
dE/dx formula is valid
Calculated range-energy curves of different R (T
): empirically determined constant
heavy particles in aluminium (num. int. of BBF): 0 min
accounting for remaining low
energy behavior of dE/dx
→ results accurate within a few percent
→ R ~ Eb
At not too high energies: -dE/dx ~ β-2 ~ T-1
T: kinetic energy
→ R~ T2
- Range energy relations are extremely useful in
Particle energy measurements
- Detector sizes
- Thickness of radiation shielding
Energy Loss of
Electrons and Positrons
• Like heavy charged particles, electrons and
positrons suffer collisional energy loss when
passing through matter
• However because of their small mass: energy
loss from emission of EM radiation caused by
scattering in the electric field of a nucleus
(bremsstrahlung, means “braking radiation”)
• Total energy loss:
 dE 
 dE 
 dE 

 =
 +

 dx  tot  dx  rad  dx  coll
•
Energy Loss of
Electrons and Positrons (cont.):
Collision Loss
Bethe-Bloch formula needs to be modified for two reasons:
1. small mass of electrons/positrons
2. for electrons the collisions are between
identical particles (in particular
maximum energy transfer becomes
Wmax=Te/2, Te: kinetic energy of incident e+, e-)
τ: kinetic energy
in units of mec2
Energy Loss of
Electrons and Positrons (cont.):
Radiation Loss: Bremsstrahlung
E < few hundred GeV, e- and e+
are the only particles for which
radiation contributes substantially
to energy loss
Why ?
Cross-section: σ ~ re2 = (e2/mc2)2
Several effects need to be considered when calculating the
energy loss:
For example screening from the atomic electrons
→ impact parameter and atomic number Z play an important
role: define screening parameter ξ ~ 1/E0Z1/3,
ξ ≈ 0: complete screening, ξ >> 1: no screening
Energy Loss of
Electrons and Positrons (cont.):
 dE 
−

 dx  coll
 dE 
−

 dx  rad
~ ln E, Z
~ E, Z2
Note: electron-electron bremstrahlung
In the field of the atomic electrons, cross-sections essentially the same as
for radiation loss in the field of the nucleus only that Z2 is replaced by Z
→ contribution can be taken into account by replacing Z2 with Z(Z+1)
in cross-section formulas
Energy Loss of
Electrons and Positrons (cont.):
Other important parameters
Critical Energy:
energy loss by radiation depends strongly on absorber
for each material a critical energy Ec can be defined as:
 dE 
 dE 
=



 dx  rad  dx  coll
for E=Ec
Approximation:
800 MeV
Ec ≅
.
Z + 12
Energy Loss of
Electrons and Positrons (cont.):
Other important parameters
Radiation length Lrad:
another quantity characteristic for the absorber,
distance over which the electron energy is reduced by a
Factor of 1/e
At high energies, where
collision loss can be ignored:
Useful approximation:
Lrad
 − x

E = E 0 exp
 Lrad 
716.4 g / cm 2 A
=
Z ( Z + 1) ln(287 / Z )
Energy Loss of
Electrons and Positrons (cont.):
Other important parameters
Critical Energy, alternative definition:
Here:
X0: radiation length
From PDG
Energy Loss of Electrons and Positrons (cont.):
Range of Electrons
Greater susceptibility to multiple scattering by nuclei
→ range of electrons is generally very different from the calculated
path length obtained from integration of dE/dx formula
Also: energy loss of electrons fluctuates much more than for heavy
charged particles (much greater energy transfer per collision, emission
of bremstrahlung)
→ great range straggeling
Energy Loss of
Electrons and Positrons (cont.):
Absorption of β Electrons
Continuous spectrum of
β decay electron energies
I = I 0 exp( − µx )
µ: β absorption constant
Number-distance curve for
Beta decay electrons from
185W
Multiple Coulomb Scattering
• Repeated elastic Coulomb scatterings of charged
particles from nuclei
• Smaller probability than inelastic collisions with
the atomic electrons
2 2
• Rutherford formula
dσ 1  Zze 
1
dΩ
Large for small angles
=


4  pv  sin 4 (θ / 2)
Multiple small angle
scatters
≈ Gaussian + Rutherford tails
Multiple Coulomb Scattering
(cont.)
Coulomb scattering distribution is well represented by the
theory of Molière
For the central 98% of the projected
angular distribution:
Define:
θ0 = θ
rms
plane
1 rms
=
θ space
2
13.6 MeV
θ0 =
z x / Lrad 1 + 0.038 ln( x / Lrad )
βcp
[
]
Backscattering
• Large-angle
deflections of
electrons along their
track
Electron-photon shower:
Shower Size
• Longitudinal development: difficult (MC)
• Transversal: Moliere radius RM=21MeV*Lrad/Ec
• Containment: 1 RM
90%
3.5 RM
99%